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Full text of "Treatise On Analysis Vol-Ii"

CHAPTER XV

NORMED ALGEBRAS AND SPECTRAL THEORY

The spectral theory of operators, which we have already encountered in an
elementary aspect in Chapter XI, is one of the masterpieces of modern
analysis. Its main object is to obtain, for linear operators on a Hilbert or
prehilbert space satisfying suitable continuity conditions, an analog of the
classical theorem of algebra which assigns a canonical form (by means of
"Jordan matrices ") to an n x n matrix over C (or, equivalently, to an endo-
morphism of a^m'te-dimensional complex vector space). In Chapter XI we
have seen how this result, suitably modified, can be extended to compact
operators. But there is another much less obvious generalization, due to
Hilbert and his successors, which applies to a wider class of operators: in
particular, to continuous self-adjoint operators (11.5), and more generally
to normal operators (15.11). Just as in the classical case a self-adjoint (or
normal) matrix over C has a canonical form which is a diagonal matrix, the
continuous normal operators can all be described in terms of a single model:
namely multiplication M^(w) :/i-> (&{/*)* *n a space LcGO by the class of an
essentially bounded function u (15.10). The Lebesgue theory intervenes here in
an essential way (even if we begin with a self-adjoint operator which comes
from a differential equation as regular as we please (Chapter XXIII)), and it
is no exaggeration to say that the occurrence of Lebesgue theory in spectral
theory and related fields such as harmonic analysis and the theory of repre-
sentations of locally compact groups is the principal reason of its importance
in Analysis.

A modern account of spectral theory does not follow the path mapped out
by Hilbert, but uses a much more elegant and powerful method based on the
theory of normed algebras inaugurated by Gelfand and his school. In this
chapter we shall concentrate mainly on normed algebras with involution
(15.4), because these are the ones which arise in spectral theory. But the
general theory of normed algebras, and especially the fundamental concepts of

304 vn (notation  of (14.11.2)) and   | un(x) df$(x) = 1. Prove that